Ch. 7 - Applications of Trigonometry and Vectors

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. 7 - Applications of Trigonometry and Vectors

Problem 14

Chapter 8, Problem 14

Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessary. See Example 1.
〈8√2, -8√2〉

Verified step by step guidance

Identify the components of the vector. Here, the vector is given as \(\langle 8\sqrt{2}, -8\sqrt{2} \rangle\), where the x-component is \(8\sqrt{2}\) and the y-component is \(-8\sqrt{2}\).

Calculate the magnitude of the vector using the formula \(\text{magnitude} = \sqrt{x^2 + y^2}\). Substitute the components: \(\sqrt{(8\sqrt{2})^2 + (-8\sqrt{2})^2}\).

Simplify the expression inside the square root by squaring each component and adding them together.

Find the direction angle \(\theta\) using the formula \(\theta = \tan^{-1}\left( \frac{y}{x} \right)\). Substitute the components: \(\theta = \tan^{-1}\left( \frac{-8\sqrt{2}}{8\sqrt{2}} \right)\).

Determine the correct quadrant for the angle based on the signs of the x and y components, then adjust the angle accordingly to find the direction angle measured counterclockwise from the positive x-axis. Round the angle to the nearest tenth of a degree.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Vector Magnitude

The magnitude of a vector represents its length and is calculated using the Pythagorean theorem. For a vector with components (x, y), the magnitude is √(x² + y²). This gives a non-negative scalar value indicating the vector's size regardless of direction.

Direction Angle of a Vector

The direction angle of a vector is the angle it makes with the positive x-axis, measured counterclockwise. It can be found using the inverse tangent function: θ = arctan(y/x). Adjustments may be needed based on the vector's quadrant to get the correct angle.

Handling Vectors with Radical Components

Vectors with components involving radicals, like √2, require careful arithmetic when calculating magnitude and direction. Simplifying expressions and using exact values before rounding helps maintain accuracy in the final results.

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Related Practice

Textbook Question

Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessary.

〈-4, 4√3〉

Textbook Question

Solve each triangle. Approximate values to the nearest tenth.

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Textbook Question

Refer to vectors a through h below. Make a copy or a sketch of each vector, and then draw a sketch to represent each of the following. For example, find a + e by placing a and e so that their initial points coincide. Then use the parallelogram rule to find the resultant, as shown in the figure on the right.

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a + (b + c)

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Textbook Question

Find the unknown angles in triangle ABC for each triangle that exists.

C = 41° 20', b = 25.9 m, c = 38.4 m

Textbook Question

Solve each triangle. Approximate values to the nearest tenth.

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Textbook Question

Solve each triangle ABC.